Wednesday, April 8 2020

2:00pm - 3:00pm

2:00pm - 3:00pm

Masters Presentation

Variance Reduction Methods Based on Multilevel Monte Carlo

ABSTRACT

If we could see into the future, then finance would be a lot easier. Unfortunately, we can’t, so stock traders work with mathematicians. When a time machine isn’t available, the next best option is a good mathematical model. While many excellent models exist to predict stock prices, their complexity often evades an analytic solution. When this is the case, simulation becomes the best alternative. What began in the Los Alamos Laboratories as Monte Carlo estimation evolved over the next 80 years to become something ubiquitous in financial mathematics. Today, Monte Carlo computational methods are so heavily used that pseudo-random numbers alone hardly suffice. Predicting the modern market requires efficiency, and to this end, a number of variance reduction techniques emerged. In this paper, I juxtapose two, and find that their combined effects are synergistic.

In 2004, Okten introduced a method of generating high quality random numbers. Somewhat paradoxically, he proposed re-randomizing the un-randomized Halton sequence. Doing so, he argued, allows us to keep the space-filling property of quasi-random sequences without the troublesome correlation between terms. In 2018, Giles published a paper that introduced the multi-level Monte Carlo (MLMC) algorithm. Instead of high-quality numbers, he sought efficiency in the structure of the algorithm. Monte Carlo algorithms naturally require averaging many estimates, but due to computational expenses, one must choose between averaging many poor estimates and averaging a few good estimates. Giles found a way to incorporate both, thus producing an even better estimate. Both methods work well by themselves, but no one has yet tried to combine them.

Chair/Advisor: Yaning Liu

Committee: Erin Austin, Burt Simon

If we could see into the future, then finance would be a lot easier. Unfortunately, we can’t, so stock traders work with mathematicians. When a time machine isn’t available, the next best option is a good mathematical model. While many excellent models exist to predict stock prices, their complexity often evades an analytic solution. When this is the case, simulation becomes the best alternative. What began in the Los Alamos Laboratories as Monte Carlo estimation evolved over the next 80 years to become something ubiquitous in financial mathematics. Today, Monte Carlo computational methods are so heavily used that pseudo-random numbers alone hardly suffice. Predicting the modern market requires efficiency, and to this end, a number of variance reduction techniques emerged. In this paper, I juxtapose two, and find that their combined effects are synergistic.

In 2004, Okten introduced a method of generating high quality random numbers. Somewhat paradoxically, he proposed re-randomizing the un-randomized Halton sequence. Doing so, he argued, allows us to keep the space-filling property of quasi-random sequences without the troublesome correlation between terms. In 2018, Giles published a paper that introduced the multi-level Monte Carlo (MLMC) algorithm. Instead of high-quality numbers, he sought efficiency in the structure of the algorithm. Monte Carlo algorithms naturally require averaging many estimates, but due to computational expenses, one must choose between averaging many poor estimates and averaging a few good estimates. Giles found a way to incorporate both, thus producing an even better estimate. Both methods work well by themselves, but no one has yet tried to combine them.

Chair/Advisor: Yaning Liu

Committee: Erin Austin, Burt Simon

Speaker: | Lu Vy |

Affiliation: | |

Location: | See Zoom link from email |

Done